I love getting myself in trouble🤣.
Skin effect is real.
You want the high pass connected for sure. The HP run has much thinner wire than the LP. There will be a clarity in the high end you just can't get with the LP section.
The LP section is designed to pass current. Woofers love current. Smearing is not an issue.
Conduct your own experiment: listen for a couple of days with only the HP connected. This will get your ears used to a certain 'sound'. Then switch to the LP only. Use music with a lot of singing. Use only well recorded songs. You will know within minutes.
If there is absolutely no difference, sell the wire. If there's a noticeable difference, now connect both and make a new observation.
Love this hobby.
I build my own speakers. 4 wires from the amp. 40awg to the tweeter, 36awg to the midrange, 18awg to the midwoofer, and 11.5awg to the woofer. 5 years of experimentation got me to this point. Love being an excessive compulsive.