Mixed odd–even period independence conjecture of Xue

For $r\ge 1$, let $K=2r+14$, let $S_K$ be the space of cusp forms of weight $K$, and put $d=\dim S_K$. The conjecture asserts that the relevant mixed odd and even Mellin-period functionals on $S_K$ in a fundamental half-range are linearly independent in the full-spark sense: every collection of $d$ distinct such period functionals is linearly independent.