Minimum clues for $5 \times n$, $6 \times n$, and $7 \times n$ Numbrix puzzles

For an $m \times n$ Numbrix puzzle, call a set of clues defining exactly one puzzle a defining clue set, and let the minimum number of clues be the smallest size of such a set. Minimum-clue conjectures. The minimum number of clues is $3$ for $5 \times n$ boards, $3$ for $6 \times n$ boards, and $4$ for $7 \times n$ boards:

$$\begin{aligned} \text{for }5 \times n &: 3,\\ \text{for }6 \times n &: 3,\\ \text{for }7 \times n &: 4. \end{aligned}$$

The authors report computer verification that two clues are insufficient for the square cases $5 \times 5$ and $6 \times 6$, and partial computational evidence for the $7 \times 7$ case; the asserted formulas for all $n$ remain conjectural.