Semi-inducibility conjecture for symmetric paths

Let $P_{E^\ell N^\ell}$ be a path of length $2\ell$ whose first $\ell$ edges are red and whose remaining $\ell$ edges are blue. Let $K(a,n)$ be the graph formed by a clique on $a$ vertices together with $n-a$ isolated vertices, and let $I(P_{E^\ell N^\ell},\beta)$ denote the maximum asymptotic density of this colored path in a red/blue clique of red density $\beta$. Semi-inducibility conjecture. For $\ell \geq 2$, $I(P_{E^\ell N^\ell},\beta)$ is asymptotically achieved by $K(a,n)$ or its complement. The case $\ell=2$ is established by the theorem preceding this conjecture, while the claim for longer symmetric paths is motivated by flag-algebra experiments and remains open.