In a 1966 paper Blakley and Dixon conjecture the following. Let $S$ be a symmetric matrix with nonnegative entries and let $u$ be a unit vector with nonnegative entries. For integers $k\ge j$ both odd,
$$\langle u, S^ku\rangle^j\ge \langle u, S^ju\rangle^k.$$
They verify this conjecture under the condition that RHS is sufficiently large. I am wondering if there has been any progress towards this conjecture.
A year earlier, Blakley and Roy verify the $j=1$ case. This also implies that the conjecture is true whenever $j$ divides $k$. There are no papers known to Google scholar citing this paper that solves the conjecture.