The Adams differentials on the classes $h_j^3$

In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes $h_j$, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill--Hopkins--Ravenel proved that the classes $h_j^2$ support non-trivial differentials for $j \geq 7$, resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes $h_j^2$ for $j \geq 7$ and the fate of $h_6^2$ remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_4$-differentials on the classes $h_j^3$ for $j \geq 6$, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory -- $\mathbb{C}$-motivic stable homotopy theory and $\mathbb{F}_2$-synthetic homotopy theory -- both in an essential way. Along the way, we also show that $h_j^2$ survives to the Adams $E_5$-page and that $h_6^2$ survives to the Adams $E_9$-page.