Conjectural Bernoulli congruences for multiple harmonic sums of depths 8, 9, and 10

Let $m=1,2$ and let $p$ be a prime with $p\ge 11$. For the multiple harmonic sums $R_d^{(m)}(p)$ and Bernoulli numbers $B_n$, the following congruences are conjectured:

Bernoulli congruences for $R_8^{(m)}(p)$, $R_9^{(m)}(p)$, and $R_{10}^{(m)}(p)$.

$$R_8^{(m)}(p)\equiv \frac{112}{5}m(m^2+16)(m^2-1) B_{p-3}B_{p-5} \pmod{p},$$ $$R_9^{(m)}(p) \equiv -\frac{8!}{18}\binom{m+2}{5} B_{p-3}^3 -8m(m^6+126m^4+1869m^2+3044) B_{p-9} \pmod{p},$$ $$R_{10}^{(m)}(p) \equiv -\frac{24}{35}m(m^4+71m^2+540)(m^2-1)\left(50B_{p-3}B_{p-7}+21B_{p-5}^2\right)\pmod{p}.$$

These formulas are conjectured for the values of $m$ and $p$ specified above. The conjecture extends the paper's established congruences for lower depths and reflects numerical evidence suggesting explicit Bernoulli-number expressions at these depths; its general validity remains unproved.