Classification conjecture for prime alternating links with crosscap number three

Let $k$ be a positive integer. For a link $L$, write $\gamma(L)$ for its crosscap number and $\Gamma(L)$ for its unoriented genus. A pretzel link $P(2\ell,2m,2n)$ is specified by positive integers satisfying $0<\ell\leq m\leq n$.

Crosscap-three classification conjecture. For each even crossing number $2k\geq 6$, there are exactly

$$p(k,3)=\left\lfloor\frac{k^2+3}{12}\right\rfloor$$

prime alternating links $L$ with $\Gamma(L)<\gamma(L)=3$, namely the pretzel links $P(2\ell,2m,2n)$ with $0<\ell\leq m\leq n$ and $\ell+m+n=k$. Here $p(k,3)$ denotes the number of partitions of $k$ into three positive parts.

The claim gives an explicit classification and count for the observed links with crosscap number exceeding unoriented genus. It is presented as a conjecture, and no resolution is supplied.