We describe a general method of proving degree lower bounds for conical juntas (nonnegative combinations of conjunctions) that compute recursively defined boolean functions. Such lower bounds are known to carry over to communication complexity. We give two applications:

$\bullet~$ $\textbf{AND-OR trees}$: We show a near-optimal $\tilde{\Omega}(n^{0.753...})$ randomised communication lower bound for the recursive NAND function (a.k.a. AND-OR tree).

$\bullet~$ $\textbf{Majority trees}$: We show an $\Omega(2.59^k)$ randomised communication lower bound for the 3-majority tree of height $k$. This improves over the state-of-the-art already in the context of randomised decision tree complexity.