Dilcher's conjecture on the higher rows of the coefficient array

Let $b_n^{(p)}$ denote the coefficients appearing in the recurrence referenced in the source, with $n,p$ positive integers. The first two rows are $b_1^{(p)}=p/12$ and $b_2^{(p)}=(5p+3)/10$. Dilcher's conjecture. The third, fourth, and fifth rows are given by

$$b_{3}^{(p)}=\frac{175p^{2}+315p+158}{140(2p+3)},$$ $$b_{4}^{(p)}=\frac{6125p^{4}+25725p^{3}+41965p^{2}+29547p+7230}{21(5p+3)(175p^{2}+315p+158)},$$

and

$$b_{5}^{(p)}=\frac{25(5p+3)(471625p^{6}+3678675p^{5}+12324235p^{4}+22096305p^{3}+22009540p^{2}+11549748p+2519472)}{132(175p^{2}+315p+158)(6125p^{4}+25725p^{3}+41965p^{2}+29547p+7230)}.$$

These formulas continue the observed pattern in the tabulated values of the higher-order Euler-polynomial coefficients; the source attributes the conjecture to Karl Dilcher, but the supplied text gives no resolution or broader general formula.