Likely #14 is low approx. 2.7 - 2.8 volts and the reason why the BMS doesn't switch.
With the other cells at 4.1v it makes sense that #14 is low, one needs to try and get a correct reading for #14.
Can you not probe directly on the cell group ends rather then the wiring ?
One can also get a cell reading via the JST connector , place a needle or pin in Red wire #14 position and also one in Red wire #13 position (don't allow pins to touch/short out). Probe both pins with a meter (again don't allow probes to short each other) one should get a correct voltage reading.
To get the BMS to work one will have to raise #14 manually to 4.1v or very close , then the BMS will switch on. No use charging to a much lower figure on #14 as the BMS won't balance a huge difference in voltage.
The reason the BMS doesn't switch is because it detects a fault , that fault being one cell group is low and below the minimum cell group voltage it expects to see.
Sometimes one can get away with 3v but it is all dependant on how the BMS is programmed.
During the non use period the BMS has bled down #14, most common fault is an open bleed resistor or the battery sw left on .
To manually charge one will need an old 5v mobile phone charger, cut the charge connector off and split the two wires. Pair back the wire ends and solder on a arduino board pin to each wire, check polarity and heat shrink a bit of Red and Black to each pin.
Then one can charge the cell group directly or via the JST sensor connector.
If the JST has 14 thin Red wires , simply put the 5v charger Red pin in the very last Red wire position and the 5v charger Black pin in the 13th Red wire position.
Switch the charger on and then carefully monitor that cell group voltage , once the group reads 4.1v or very near try charging as normal.