Periodicity of coexponents of h_0-submodules in the v_1-periodic algebra

Let $j\geq 3$ be fixed, and for each integer $i$ consider the $h_0$-submodule cogenerated by

$$\frac{\overline{v_1^{3\cdot 2^j+i\cdot 2^{j+1}}}}{h_0}.$$

Periodicity conjecture. These $h_0$-submodules have the same $h_0$-coexponent for all $i$.

This conjecture asserts uniformity of the coexponents along the indicated $v_1$-periodic families in the classical Adams $E_2$-page. The supplied text presents the claim as suggested by computed divisibility data, but gives no evidence of a proof or disproof.