Conjectural twisted Stong calculation for connective twisted KO-theory

Let $k_2o\langle r\rangle$ denote the indicated connective twisted $KO$-theory spectrum, let $\underline{\mathcal A(1)}$ be the twisted Steenrod subalgebra, let $\varphi$ be the embedding of $\mathcal A(1)$ into the twisted Steenrod algebra, and write $sh^m$ for grading shift. Twisted Stong calculation conjecture. As $\underline{\mathcal A(1)}$-modules, one has

$$sh^{-(8n+1)}\underline{\mathcal A(1)}/\varphi(Sq^2)\cong H^*(k_2o\langle 8n+1\rangle),$$

and

$$sh^{-(8n+4)}\underline{\mathcal A(1)}/\varphi(Sq^{1,5})\cong H^*(k_2o\langle 8n+4\rangle).$$

The claim is motivated by Stong's computation for ordinary connective $KO$-theory and is presented only as a natural guess; no proof or resolution is supplied.