Generalized Catalan derivative formula for unit-step solutions

Let $p=1,2,\dots$, let $j=1,\dots,p+1$, and let $n=0,1,\dots$. For the functions $f_j$ corresponding to initial data in the form of the unit step, let $C^p_{j,n}$ denote the generalized Catalan numbers. Generalized Catalan derivative conjecture. The derivatives satisfy

$$f^{(n)}_j(0)=C^p_{j,n}.$$

Equivalently, $f_1,\dots,f_{p+1}$ are the exponential generating functions for the generalized Catalan numbers. The statement identifies the Taylor coefficients of the unit-step solutions of the Bogoyavlensky lattice with generalized Catalan numbers; the supplied text gives no evidence establishing or disproving it.