Construction 2 weight-spectrum conjecture for Reed–Muller codes

Let $S$ be the set of all Hamming weights generated by Construction 2, namely the construction producing functions $g=0||(g_1+a_1)||g_2||(g_1+g_2+a_2)\in RM(m+1,2m+2)$. Here $a_1,a_2\in F_2$, and the parameters satisfy $m\ge 4$.

Construction 2 weight-spectrum conjecture. The set $S$ contains all weights in the two families

$$\{2^{m+2}+2^m+2i\}\cup\{2^{2m}+2^m+2i\},$$

where $0\le i<2^m$.

This conjecture is proposed after analyzing the weights produced by Construction 2 for $m=4,5$, and the source states that it implies the preceding weight-spectrum conjecture. No resolution is given in the supplied text.