Delsarte problem for 4-designs on the unit 3-sphere

The extremal Delsarte problem $A(d,s)$ for spherical $s$-designs allows us to estimate from below the minimum number of nodes $N(d,s)$ of a weighted quadrature formula on the sphere $\mathbb{S}^{d}$. We prove that
$$ A(3,4)=14.560317967882\ldots. $$
Hence $N(3,4)\ge 15$. Our open conjecture is that $N(3,4)=16$.