From a cursory reading of a summary by Kortlandt (either in The expansion of the Indo-European languages, 2018, or something similar), I gained the impression that the two families were initially unrelated but later came into contact. That's my own impression, I don't mean to misrepresent the author. He is noted as Indo-Uralic author in Wikipedia's respective category--which doesn't mean much. He seems very careful not to submit to any premature judgement, but certainly shows interest in the comparison.
@Annix submitted under another question that proto-uralic had no number words, and took these from IE's.
@... (Midas?) has an unanswered question here asking for evidence of Uralic loans into Indo-Iranian (or later); Loans into the opposite direction are known. This probably has something to do with assuming that Indo-Iranians moved from the Steppe westwards around the Caspian--which is an unsettled matter of debate. Kortlandt also references indications that Uralic folk moved westwards into the Steppes "at the same time". Personally I'm not giving this two much weight for two reasons: a) The single Mother-Hypothesis is untennable, one dimensional thinking is but a necessary simplification--look for example at the account of Kurdish language in the Encyclopedia Iranica for a good deal of assymetrie b) The Steppe Hypothesis is convincing, but not the last word, because, how, while presence of artifacts is certain, Urheimat should be more than a nebulous idea, I don't know; Also, recent results (cf Gamkrelidze, ejectives, 2010) again favour early South Carpathian presence drawing on DNA analysis--which Kortlandt did not mention, iirc--of course, one shouldn't confuse Language with Genetics (incidentally, Kortlandt has written on the mathematical side of things, too, which I just discovered writing this question).
As an aside, you should ponder the question, if a child can come from two parents.
@Curiousdannii is correct in the comments, insofar any proof must imply that a proto language can be reconstructed, if you understand relation as single origin. However, since you seem to be aware that no such reconstruction has been achieved--and to be fair let's say that I have yet not read past the intro of "A Storm of Words", that wad linked by @Rock, and I would not understand half of it, but have to rely on the approximate judgement, in which I follow @Draconis' reservations--you are rather asking about methodology, and whether a proof can be constructed by other means.
Ultimately, it is programatic. Given the necessary difficulties that had to be expected, if the hypothesis were true in one form or other weaker ones, wonder, are enough people working on it to find out?
!image, now spread all over the web, originally from the guardian, I believe
You might as well wonder for the time being, in an absurd sense, when you look at your own family tree and how it seems to grow exponentially along the time axis, considering that twohundred generations back saw only ten 10, or perhaps 100 billion people on the planet, whether your parents were siblings and if you can reconstruct Askr and Embla (or Adam and Eve) from that.
