Surjectivity conjecture for sequential and spin CNOT layouts

For $n$ qubits, let $\mathsf{sequ}(L)$ denote the layout consisting of $L$ CNOT units in sequence, and let $\mathsf{spin}(L)$ denote the alternating spin layout. Write $V_{\mathsf{sequ}(L)}$ and $V_{\mathsf{spin}(L)}$ for their corresponding circuit-to-unitary maps. Surjectivity conjecture. The two maps are surjective whenever

$$L\geq\frac{1}{4}(4^n-3n-1),$$

which is the TLB. The claim concerns whether these repeating layouts can realize all target unitaries; the supplied text says that experimental results support it, but does not provide a proof or resolution.