Unique Nash-equilibria in multi-unit auctions with uncertain participation

Setup
Consider a one shot sealed bid multi-unit auction where $$N$$ bidders compete for $$K$$ identical objects and each bidder $$i$$ has demand $$d_i\in \{1,\dots,K\}$$. Bidders receive private i.i.d. signals on the value of the objects $$v_i^1, \ldots, v_i^{d_i}$$. The pricing rule can be either pay-as-bid or uniform price with the clearing price equal the bid of the last accepted bidder.

Assume further that participation is uncertain: Any given player is inactive with positive (but small) probability. Inactivity is independent across bidders and of the signals.

Question(s)
Is there a unique (Bayes-)Nash-equilibrium?
If so, under what conditions (pricing rule, distributions for private values, ...)?

I am looking for literature which either provides an argument for unique equilibria or for an example with multiple equilibria.

I conjecture (based on Ausubel et. al 2014) that one cannot prove uniqueness of equilibria independent of the pricing rule.