HLS conjecture for Heegner divisors of Coble type quartics

Let $d4a4_2$, $d4a4_6$, $d4a4_8$, and $d4a4_d$ for $de5 16$ denote the indicated Heegner divisors, and call a Heegner divisor HLS if it has the HLS property used in the paper. HLS conjecture. The Heegner divisors $d4a4_2$, $d4a4_6$, and $d4a4_8$ are also HLS divisors, whereas the Heegner divisors $d4a4_d$ for $de5 16$ are not HLS. The first assertion is expected to follow from the examples discussed in the paper, while the second is supported by a special example; the claims remain conjectural.