A direct-sum theorem for the non-deterministic communication complexity of inequality?
I think this can be done with $\log n \cdot\log t + t \cdot\log\log t$ bits of nondeterministic communication.

By encoding all strings with an error correcting code, we may assume that whenever we have an input pair $(x,y)$ with $x\neq y$ the two strings differ in a constant fraction of coordinates. Suppose we are given $t$ pairs $(x^j,y^j)_{j\in[t]}$ that we want to certify are all nonequal. By averaging, there is some coordinate $i\in[n]$ that witnesses $x^j_i\neq y^j_i$ for a constant fraction of $j\in[t]$. Iterating this argument we may construct a set $S\subseteq[n]$ of about $|S|=\log t$ coordinates such that for every $j$ there is some $i\in S$ such that $x^j_i\neq y^j_i$. The nondeterministic proof consists of $S$ (which is $\log t\cdot \log n$ bits) and a mapping $[t]\to S$ (which is $t\cdot\log\log t$ bits) that indicates which coordinate in $S$ to use for each of the $t$ input pairs.