The Catalan-array conjecture for vertically-recurrent matrices

Let $C_n$ denote the Catalan triangular array. A matrix is vertically recurrent when it satisfies the paper's vertical recurrence relation. The Catalan-array conjecture. The Catalan array $C_n$ is a vertically-recurrent matrix. This conjecture asserts that the Catalan array belongs to the class of vertically-recurrent matrices; the supplied text gives no resolution or supporting result.