Odd-weight alternating odd-position conjecture for block decompositions

Let $\mathcal O=\{\ell_1,\ell_3,\ldots,\ell_{2n-1}\}$ and $\mathcal E=\{\ell_2,\ell_4,\ldots,\ell_{2n}\}$. For each $i$, let $\mathcal E_i$ be $\mathcal E$ with $\ell_{2i}$ omitted, and let $x$ satisfy $x+\sum_{\ell_{2j+1}\in\mathcal O}\ell_{2j+1}$ odd and $x-\sum_{\ell_{2j}\in\mathcal E_i}\ell_{2j}>0$. Construct $\mathcal R_i$ by interleaving $\mathcal O$ and $\mathcal E_i$, inserting the latter difference immediately before and immediately after $\ell_{2i-1}$, and alternating over $\mathcal O$:

$$\mathcal R_i=\operatorname{Alt}_{\mathcal O}\bigl(I_{\mathrm{bl}}(\mathcal B_i)+I_{\mathrm{bl}}(\mathcal C_i)\bigr).$$

Odd-weight alternate odd-position conjecture. Subject to further restrictions on the block lengths and $x$,

$$\sum_{i=1}^{2n}(-1)^i\mathcal R_i=0.$$

Every term has odd total weight $x+\sum_{\ell_{2j+1}\in\mathcal O}\ell_{2j+1}$. The source presents this as a candidate with unspecified additional restrictions, so its precise domain remains open.