Kirillov's lower-bound conjectures for small-field unitriangular groups

For $n\geq 1$, let $U_n(q)$ be the group of upper triangular $n\times n$ matrices over $\mathbb F_q$ with ones on the diagonal, and let $k(U_n(q))$ denote its number of conjugacy classes. Let $\{A_n\}$ be the Euler sequence and $\{B_n\}$ the Springer sequence. Kirillov's conjectures. For all $n\geq 1$,

$$k\bigl(U_n(2)\bigr)\geq A_{n+1}$$

and

$$k\bigl(U_n(3)\bigr)\geq B_{n+1}.$$

These lower bounds are attributed to Kirillov; the source does not give their resolution.