Thanks for your patience on my slow answer to this (and also on your SVD PR - we really do need to get to that and appreciate your contribution).
So I haven't thought about the non-contracting product in the context of QN's for a while, but I just thought about it a bit and I think recall what the problem was. (I'll give a more concrete example below.) The problem, I believe, is that there are lots of situations where if you input two QN-conserving ITensors, which currently we require to have a single, well-defined overall QN flux, then the output of the non-contracting product can result in a tensor that violates this condition, and will not have a well-defined QN flux.
There may be some resolutions to this issue:
1. we could just throw an error for these cases, but still allow the non-contracting product to work otherwise
2. we could introduce a more general kind of QN ITensor which is allowed to have non-zero blocks corresponding to *different* fluxes
However, I'm not sure if 1 is worth it, and 2 opens up a whole can of worms in terms of what guarantees we can provide about QN conservation and also about performance of block-sparse calculations (i.e. what if the quote "block sparse" tensor has most of its blocks actually non-zero?).
Now for a concrete example (and please tell me if you happen to catch a flaw in this example, as I may have misremembered some details even about what non-contracting product is supposed to do), say there are two tensors A and B. Let's say they each individually have a well-defined QN flux of zero. Let's say they each have two indices as well. A has indices i and j, and B has indices j and k (all with Out arrows). So that means that any non-zero block of A or B correspond to settings of the indices where the QN's of i and j add up to zero, or the QNs of j and k add up to zero.
Now if we do the non-contracting product of A and B, what we mean is that we "lock" or "tie" the j index of each together, to get a new tensor C_ijk. But when we do this, say then if Index i carries a QN q, then j will have to carry a QN of -q by the condition that A has zero flux. But then also k will carry a QN of q because B also had zero flux. So the overall QN of that block will be q-q+q which is q. However q can take multiple values, so then the flux of C can take multiple values, making it not a valid QN ITensor.
So I hope that example is both clear and accurate!