Tensor-product Fourier dual-certificate conjecture for Hadamard matrices

Let $U(d)$ be the unitary group, let $\nu$ be normalized Haar measure on $U(d)$, let $e$ be its identity, and let $\mu_{[\bm H]}$ be the smoothed measure associated with the equivalence class of a Hadamard matrix $\bm H\in U(d)$. Dual-certificate conjecture. For every $d$, there exists a Hadamard matrix $\bm H\in U(d)$ such that

$$\delta_e+d\mu_{[\bm H]}\succeq(d+1)\nu,$$

or equivalently, $\widehat{\mu_{[\bm H]}}(\pi)\succeq-\frac1d\bm I$ for every irreducible representation $\pi$ of $U(d)$. If $d=p_1^{k_1}\cdots p_m^{k_m}$ is the prime factorization of $d$, one can take

$$\bm H=\bigotimes_{k=1}^m\bm F_{p_k}^{\otimes k_k}.$$

Such a certificate would imply the conjectured dimension-independent lower bound $B(d)=d+1$. The construction is motivated by observed tensor-product patterns in known mutually unbiased bases, but its validity remains to be verified.