Necessity of the zero-pattern condition for surjective DQSOs

Let $S$ be the simplex on which a discrete quadratic stochastic operator (DQSO) $V$ acts, and write its coefficients as $P_{ij,k}$. Suppose that there is a sequence j_n_{n\geq1}\subset\mathbb N satisfying

$$P_{j_nj_m,k}=0,\qquad \forall k\notin\{n,m\}.$$

Necessity conjecture. If $V$ is surjective on $S$, then there exists a sequence j_n_{n\geq1}\subset\mathbb N such that

$$P_{j_nj_m,k}=0,\qquad \forall k\notin\{n,m\}.$$

The displayed condition is known to be sufficient for surjectivity, but the paper does not prove its necessity; the conjecture is motivated by the examples constructed there.