Planar Veronese secant-index formula

Fix $d>0$ and set $N=N_d^{(2)}$. Define the sequence $(a_1,\ldots,a_{N-3})$ by the displayed pattern in the source, and define $(b_1,\ldots,b_{N-2})$ by

$$b_j=d^n-\sum_{i=1}^{j-1}a_i.$$

Planar Veronese secant-index conjecture. The maximal number of points in the finite reduced intersection of $r$ linearly independent degree-$d$ curves is $b_{r-1}$. The statement proposes an explicit continuation of the observed planar Veronese pattern, but the source does not establish it.