Total positivity of Fuss–Narayana triangles and strict positivity of their squares

For an integer $m\geq 1$ and $n\geq k\geq 0$, define

$$FNA_{\langle m\rangle}(n,k)=\frac{1}{n+1}\binom{n+1}{k}\binom{m(n+1)}{n-k},\qquad FNB_{\langle m\rangle}(n,k)=\binom{n}{k}\binom{mn}{n-k}.$$

Let $FN_{A,\langle m\rangle}$ and $\overleftarrow{FN}_{A,\langle m\rangle}$ be the corresponding type-$A$ Fuss–Narayana triangles, and let $FN_{B,\langle m\rangle}$ and $\overleftarrow{FN}_{B,\langle m\rangle}$ be the corresponding type-$B$ triangles; define the associated Fuss–Narayana squares by

$$FN_{A,\langle m\rangle}^{\ulcorner}=\left[FNA_{\langle m\rangle}(n+k,k)\right]_{n,k\geq 0},\qquad FN_{B,\langle m\rangle}^{\ulcorner}=\left[FNB_{\langle m\rangle}(n+k,k)\right]_{n,k\geq 0}.$$

Fuss–Narayana conjecture. For any $m\geq 1$, the Fuss–Narayana triangles are totally positive and the Fuss–Narayana squares are strictly totally positive. The source introduces this as a proposed conjecture, and no resolution is supplied.