Dimension-independent Welch-bound conjecture for positive definite functions

Let $U(d)$ be the unitary group, let $H(d)$ be the set of complex Hadamard matrices in $U(d)$, and let $B(d)$ be the optimal positive-definite-functions upper bound for mutually unbiased bases, with $B(d)\leq d+1$. Strong lower-bound conjecture. For every integer $d\geq 1$, the method of positive definite functions cannot improve on the Welch bound, so that $B(d)=d+1$. This would rule out any improvement by this method in every dimension. The paper proves the corresponding polynomial bound $B_{\leq 6}(6)=7$ but does not establish the assertion for all dimensions.