I don't have access to hard data or facts, so I can only offer you anecdotal observations gleaned from my last 20 years in IT.
I believe there is a vast difference between the that way most developers create software today as compared to 20 years ago. With the Agile movement having gained so much momentum, particularly in the last 5-6 years, I've seen a real shift in attitudes in the workplace. So much so that the quality of what we do just seems to grow in leaps and bounds every year, and with every project as we apply the lessons we've learned from project to project. Leaner processes combined with a focus on test-first development has grown from being very controversial to commonplace. So much so that to walk into many companies today, if you are not comfortable with Agile you'll be lucky if they don't show you the door.
So what impact has this had. First of all, I have noticed that problems do often get identified much earlier. Often it is the case that if the problem doesn't look to be too great, it can sometimes be put on hold indefinitely. In a rare handful of cases, I have seen bugs that were thought to be trivial become serious problems when addressed later, as some fundamental issue becomes apparent that wasn't considered at the time. Sometimes this can lead to a drawn out fix cycle, and that can be costly to a degree, but that cost is often measured less in terms of resourcing, and more often in terms of the impact on the relationship between customer and developer. Customers are growing used to this Agile way of thinking, which returns results to them much faster than it did in the old days, with highly iterative development sprints and fast turnaround between requests and implementation, so they have come to expect a great deal of us. And as far as the actual bugs are concerned, the time to get a bug fixed is more often greatly diminished as a result of having a solid suite of tests to support changes, and the ability to create new tests from which to provide insight and solutions to the problems reported.
So on the whole, it appears that the overall effort to fix bugs has been in most cases reduced if there is a robust suite of tests in place, and procedures to ensure that testing remains the focus of what the developer does, but the actual cost has in some ways shifted in part at least from the implementation, to other areas of the business, because in some ways, the focus has also shifted from pure supply and demand to relationship management.
Another thing that has become apparent, is that our gut instincts of a few years ago which suggested that being Agile would reduce our maintenance cycles has been proven to a degree both right and wrong. Right in the sense that solid testing has made it easier to debug and fix our code to a large degree, and to reduce overall the number of bugs released into production code, and wrong in the sense that we are now working harder to avoid needing to maintain legacy code, by constantly refactoring code and improving architecture such that it is becoming rarer that we need to develop new products completely from scratch.
So in the end, what does this mean with regards to the OP's question? Well, it means that the answer really isn't as cut-and-dry as we once might have thought it to be. 15 years ago, I would have probably answered the question as a Yes, but now I feel it's more realistic to say that it really is too difficult to measure empirically, because the nature of what we do to develop software has changed greatly from when we first started asking ourselves the OP's question back then. In some ways, the more we advance our techniques and skills as an industry, the further the question grows from a definitive yes, to a point where I suspect that in a short number of years we'll be saying that it doesn't matter when we fix bugs, because our tests and processes will be so much more robust, that the timing of the bug fixes will be less driven by efforts to save our budgets, and more by priorities to satisfy our customers needs, and the relative cost will become virtually meaningless contextually.
But as I say, this isn't hard data-supported evidence, just my observations of the past few years, and my gut telling me that there will be more ground-shaking wisdom to come that will improve the way that we do things.