Possible tile numbers for space-efficient 7-mosaics with no empty rows or columns

Let $K$ be a prime knot and let a space-efficient 7-mosaic of $K$ be a 7-mosaic whose tile number is minimal among mosaics of $K$. Suppose that every column or every row of the mosaic is occupied. Possible tile-number conjecture. The tile number of the mosaic can only be one of

$$27, 29, 31, 32, 34, 36, 37, 39, 40, 41.$$

This restricts the tile numbers that can occur for the specified class of prime-knot 7-mosaics; the source provides no resolution status or further context for whether all listed values occur.