Nonvanishing conjecture for symmetric Wigner 6j symbols

Let $k,n$ be integers and let

$$\left\{\begin{array}{ccc}k&k&k\\frac n2&\frac n2&\frac n2\end{array}\right\}$$

be the Wigner $6j$ symbol appearing in the description of the cubic invariant $\mathscr{P}_{n,3}(F)$. Nonvanishing conjecture. For every pair of integers $(k,n)$ with $n\ge k\ge2$, except $(k,n)=(2,3)$, one has

$$\left\{\begin{array}{ccc}k&k&k\\frac n2&\frac n2&\frac n2\end{array}\right\}\ne0.$$

The claim is a purely combinatorial conjecture motivated by computer checks and by the proposed regular-sequence extension of Dixmier's conjecture; the source does not state that it has been proved or disproved.