Total positivity conjecture for the Narayana triangle

Let $N=[N(n,k)]_{n,k\ge 1}$ be the Narayana triangle, with entries

$$N(n,k)=\frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1}.$$

Total positivity conjecture. The Narayana triangle $N$ is totally positive: every minor of $N$ is nonnegative.

The Narayana triangle is also called the Catalan triangle in the source because its row sums are the Catalan numbers. The supplied text gives no resolution status.