Conjecture on the longest increasing subsequence of conjugation-invariant permutations

Let $(\tau_n)_n$ be a sequence of conjugation-invariant random permutations, where $\tau_n$ has size $n$ and random cycle type $t^{(n)}$. Suppose that

$$t_1^{(n)}/\sqrt{n}\longrightarrow0$$

in probability. Conjecture on the longest increasing subsequence. Then

$$\frac{1}{\sqrt{n}}\operatorname{LIS}(\tau_n)\underset{n\to\infty}{\longrightarrow}2$$

in probability. The result would identify the same first-order asymptotic as for uniform permutations when the number of fixed points is negligible compared with $\sqrt n$; related cases are known for random involutions and under additional cycle constraints.