Chen's conjectures on inequalities for the Andrews spt-function

Let $p(n)$ denote the partition function and let ${\mathrm{spt}}(n)$ denote the smallest-parts function, which counts the smallest parts among the partitions of $n$. Chen's conjectures. The following inequalities hold:

1. For $n\geq 5$,

$$\frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<{\mathrm{spt}}(n)<\sqrt{n}\,p(n).$$

2. For $(a,b)\neq(2,2)$ or $(3,3)$,

$${\mathrm{spt}}(a)\,{\mathrm{spt}}(b)>{\mathrm{spt}}(a+b).$$

3. For $n\geq 36$,

$${\mathrm{spt}}(n)^2>{\mathrm{spt}}(n-1)\,{\mathrm{spt}}(n+1).$$

4. For $n>m>1$,

$${\mathrm{spt}}(n)^2>{\mathrm{spt}}(n-m)\,{\mathrm{spt}}(n+m).$$

5. For $n\geq 13$,

$$\frac{{\mathrm{spt}}(n-1)}{{\mathrm{spt}}(n)}\left(1+\frac{1}{n}\right)>\frac{{\mathrm{spt}}(n)}{{\mathrm{spt}}(n+1)}.$$

6. For $n\geq 73$,

$$\frac{{\mathrm{spt}}(n-1)}{{\mathrm{spt}}(n)}\left(1+\frac{\pi}{\sqrt{24}n^{3/2}}\right)>\frac{{\mathrm{spt}}(n)}{{\mathrm{spt}}(n+1)}.$$

These inequalities concern lower and upper bounds, supermultiplicativity, log-concavity, and ratio inequalities for the spt-function. The paper introduces them as recent conjectures of Chen; the supplied text does not indicate that any of them has been resolved.