Principality conjecture for algebraic-principal Heegner cycles

Let $\mathrm{Pin}^{\mathrm{alg}}$ be the subgroup of algebraic principal cycles and let $\mathrm{Pin}$ be the subgroup of principal cycles in the group of Heegner cycles. Principality conjecture. Every algebraic principal cycle is principal, namely

$$\mathrm{Pin}^{\mathrm{alg}}\subseteq\mathrm{Pin}.$$

If true, this would upgrade the proved modularity result with coefficients in the algebraic Heegner-cycle quotient to the corresponding result in the full Heegner-cycle quotient.