Another way of saying this is that in Zcash, the content of a fully shielded transaction does not give an adversary any more information about the possible input distribution than they could guess without seeing the content (i.e. only based on the timestamp and the number of JoinSplits in that transaction). In Monero, the adversary can refine their guess of the distribution based on the inputs that are actually mixed in, and that is what creates the privacy weakness.
Figure 8 does not apply to Zcash, it is specific to Monero, as the caption states.
-- Daira Hopwood (Zcash developer)
> In Monero, the adversary can refine their guess of the distribution based on the inputs that are actually mixed in, and that is what creates the privacy weakness.
That is not what is claimed in Section 4 of the paper. Section 4 merely indicates that of potential outputs, the time distribution introduces a bias toward the most recent (actually in Monero this might be inaccurate in some cases too: very, very recent might be less likely than merely very recent; the paper does not examine this). In Zerocash the same time distribution bias exists, though across a larger set of potential coins (or notes or whatever it is you call it).
However, very old members of that set are essentially irrelevant as their probability in the distribution is almost certainly extremely low (this is the same reason that more older outputs in Monero are essentially irrelevant).
It's the same claim as for semantically secure encryption, for example: no competent cryptographer would claim that encrypting a message implies that the adversary's knowledge of the plaintext distribution is uniform; only that the ciphertext gives the attacker no further information (apart from length, typically) about the distribution.
It is, in the same sense that the first order anonymity set of Monero transactions is all outputs included in the ring signature which can't be proven implausible (e.g. using the methods in Section 3 of the paper). However, Section 4 of the paper points out that a non-uniform distribution means this is reduced, in practice, to a smaller effective degree. The same method can be used with Zcash to estimate a smaller effective degree since many previous shielded transactions are probabilistically unlikely.
This is certainly not 'deanonymization' or 'tracing' or any such thing, but it isn't that in the Monero case either.
This way of stating is somewhat questionable in light of the claims in the second half of the paper.
What is shown in the second half of the paper is that all possible sources are not equally likely and this most probably applies to Zcash (and every other coin) as well. In the Figure 1 illustration of Zcash, it is most likely that the rightmost (most recent) arc is the correct one. Of course this can't be stated with certainty in either coin.
Another way of interpreting the trend shown in Figure 8 is that Zcash gains little (though of course it still gains something) from including all transactions in the anonymity set (arbitrarily far to the right) because once one departs from focusing predominantly on the more recent transactions, the effective anonymity set does not grow much.
> Instead, it looks like clients weren't even doing basic checks:
>> We find that among Monero transaction inputs with one or more mixins, 62% of these are deducible, i.e. they can be incontrovertibly linked to the prior TXO they spend.
There are no basic checks that can solve that issue. It was fixed in a different way.
> even if Monero fixes everything in this upcoming release,
Most of the issues in the paper were already addressed in the past, and the paper says this. The remaining issue is the time bias which the paper states has already been improved, but can be improved further.