The conjecture on the local maxima of the inducibility profile of $K_{2,2,1}$

Let $I_{K_{2,2,1}}(e)$ denote the inducibility profile of the complete $3$-partite graph $K_{2,2,1}$. For $k\geq 3$, set

$$a=\frac{k+\sqrt{(1-e)k^2-ek}}{k^2+k}.$$

The conjectured profile of $K_{2,2,1}$. The function is conjectured to satisfy

$$I_{K_{2,2,1}}(e)=\begin{cases} \sqrt{25/24}\cdot e^{2.5} & \text{if }e\in[0,2/3],\\[6pt] [15a^5(k-2)+30(ak-1)^2a^3-15(ak-1)a^4](k-1)k & \text{if }e\in\left[\frac{k-1}{k},\frac{k}{k+1}\right]. \end{cases}$$

The proposed description concerns the shape of the inducibility profile, which is continuous but appears not to be differentiable at edge densities of the form $(k-1)/k$. Experiments with other multipartite graphs suggest that sharper peaks may produce additional local maxima.