Mosaic number conjecture for the knots $6_{1}$ and $6_{3}$

For a positive integer $n$, an $n$-mosaic is an $n\times n$ mosaic-tile board representing a link or knot, and the mosaic number of a knot is the smallest natural number $m$ for which it has a knot $m$-mosaic. The knots $6_{1}$ and $6_{3}$ denote the standard six-crossing knots in knot tables. Mosaic number conjecture for $6_{1}$ and $6_{3}$.

$$\text{The mosaic number of }6_{1}\text{ and }6_{3}\text{ is }6.$$

The claim concerns the unresolved determination of mosaic numbers for knots with at most ten crossings; the source gives no evidence of a resolution.