Sums of four polygonal numbers with coefficients

Let $m\ge 3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\left(\genfrac{}{}{0px}{}{n}{2}\right)+n$ $(n=0,1,2,\dots )$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers of order $m+2$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1)+p_{m+2}(x_2)+a{p}_{m+2}(x_3)+b{p}_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\in \{(1,3),(2,4)\}$, or $(a,b)=(1,1)\mathrm{\&}4\mid m$, or $(a,b)=(2,2)\ &
m\not\equiv 2\pmod 4$. In particular, we confirm a conjecture of Z.-W. Sun which states that any natural number can be written as $p_6(x_1) + p_6(x_2) + 2p_6(x_3) + 4p_6(x_4)$with$x_1,x_2,x_3,x_4$ nonnegative integers.