The finitely forcible solution conjecture for graphon constraints

Let $F_1,\dots,F_k$ be finite graphs, let $a_1,\dots,a_k$ be prescribed values, and let $t(F_i,W)$ denote the homomorphism density of $F_i$ in a graphon $W$. Finitely forcible solution conjecture. If a finite set of constraints

$$t(F_i,W)=a_i\qquad (i=1,\dots,k)$$

is satisfied by some graphon, then it is satisfied by a finitely forcible graphon. This would imply that every extremal graph problem has a finitely forcible solution.