conjectures range on a spectrum from formal to informal. for example Hilberts famous conjecture about the decidability of mathematics was formalized into a few problems eg Hilberts 10th problem but it was also a more grandiose informal conjecture spanning the whole field. it can also be seen as a proposed research program.
one easy recipe to find such "obituary of dead conjectures" would be to consider the "meta-" statement "[x] conjecture could be proved in my lifetime." mathematics literature is full of such statements/ expectations that turned out to be "false" in the sense of utterly defying expectations about difficulty and accessibility of a proof. a classic one is the Riemann conjecture, open for over ~1½ century. applying this same model to complexity theory is not as easy because complexity theory is a much younger scientific field. however, heres a key example.
the early discovery of the P vs NP problem (now open 4½ decades) had a sort of innocence in that the original investigators did not and could not have imagined how hard or crosscutting the problem would turn out to be. to make this more specific, consider the field of circuit complexity invented in the early 1980s eg by Sipser. this was a research program somewhat like Hilberts mounted in part to attack P vs NP. some of the historical outcome is summarized by Arvind in this abstract/ introduction The computational Complexity Column, BEATCS 106:
The 1980’s was a golden period for Boolean circuit complexity lower bounds.
There were major breakthroughs. For example, Razborov’s exponential size
lower bound for monotone Boolean circuits computing the Clique function and
the Razborov-Smolensky superpolynomial size lower bounds for constant depth
circuits with MODp gates for prime p. These results made researchers
optimistic of progress on big lower bound questions and complexity class separations.
However, in the last two decades, this optimism gradually turned into
despair. We still do not know how to prove superpolynomial lower bounds for
constant-depth circuits with MOD6 gates for a function computable in exponential
there were two key papers that shot down hopes in the field. Razborov had great/ celebrated results on the Clique function but then wrote two opposing papers. one paper showed that Matching, a P-time problem, requires exponential monotone circuits and therefore in some sense the monotone circuit approach to lower bounds was thwarted because of a lack of correspondence in complexity with nonmonotone ("complete") circuits (still not fully understood).
this was expanded on in his famous paper Natural Proofs coauthored with Rudich in which it is shown that all prior circuit lower bounds proofs are subject to a particular pattern which has provable weakness in the sense of conflicting with conjectured lower bounds on hard random number generators from cryptography.
so, to some degree circuits have "fallen from grace". it is still a massive research area but the conventional wisdom, supported by technical results, is that some kind of special as-yet-unknown proof pattern/ structure would be required to get strong results in the area, if actually even possible. in fact similarly one might suggest that even "strong lower bounds in complexity theory" overall are now seen to be extremely difficult, and this was not widely expected/ predicted in the younger days of the field. but on the other hand this then ranks them up there in difficulty/ significance/ importance with the big (open) problems of mathematics.