Let $k\ge 4$. Is it true that$$\mathrm{ex}(n;H_k){\ll }_k{n}^{3/2},$$where $H_k$ is the graph on vertices $x,y_1,\dots ,y_k,z_1,\dots ,z_{\left(\genfrac{}{}{0ex}{}{k}{2}\right)}$, where $x$ is adjacent to all $y_i$ and each pair of $y_i,y_j$ is adjacent to a unique $z_i$.