Uniqueness of equilibria in first-price auction with discrete valuations

Assume we have a first-price auction with discrete and independent private values and identical distributions of all bidders. It is known that there exists a Bayes-Nash equilibrium where bidders bid on disjunct bid intervals. For example, we have valuations 0,1,2. Then a bidder with valuation 0 bids zero, a bidder with valuation 1 plays a mixed strategy on some interval $[0,overline{b}_1]$ and a bidder with valuation 2 plays a mixed strategy on some interval $[overline{b}_1,overline{b}_2]$.

Is it true that this Bayes-Nash equilibrium is unique? If this is true, how one can show that overlapping bidding intervals are impossible? Or is it possible to show at least that this is the unique symmetric equilibrium, i.e. an equilibrium where all bidders employ the same strategy?

I tried to find the answer in the literature but I only found papers which derive the symmetric Bayes-Nash equilibrium with disjunct bidding intervals without discussing other possible equilibria. Thank you!