Edge inducibility conjecture for odd paths

Let $P_k$ be a path on $k$ vertices, let $c(G,P_k)$ be the number of induced copies of $P_k$ in a graph $G$, and define

$$\rho(P_k,m)=\max\{c(G,P_k)\mid |E(G)|=m\}.$$

Edge inducibility conjecture for odd paths. If $k\geq 5$ is odd, then

$$\rho(P_k,m)=(1+o(1))4\left(\frac{m}{k+1}\right)^{(k+1)/2}.$$

The construction preceding the conjecture gives this quantity as an asymptotic lower bound via an unbalanced blow-up of $P_k$, but the source does not establish the matching upper bound.