As stated in the answer which you linked to:
[I]t is true that the Shulhan Arukh was mahmir and brought the shittah of the Semag that since roofs in his time were made out of such planed boards, one should not use boards for sekhakh, lest it be seen as a roof...
So the starting point for not using planks, even when they are narrower than four tefachim, is the custom brought in Shulchan Aruch Orach Chaim 629:18:
ואם אין ברחבן ד' כשרים ... ונהגו שלא לסכך בהם כלל
If their width is less than four tefachim they are valid ... And our custom is not to use them as sechach at all.
And see Mishnah Berurah ad loc. who provides two reasons for this custom:
- That sechach made of such planks is too watertight
- That people use planks narrower than four tefachim for roofing
Now, how narrow can a plank be before it is permitted for use?
Rav E. Melamed summarizes in his Harchavot Lifninei Halachah, Sukkot, 2:4:4:
אמנם לא כתבו עד איזה רוחב צריך להחמיר. אבל מוסכם שמותר לסכך בנסרים שרוחבם פחות מטפח, וכן נהגו להורות בירושלים ... בפועל, אין כיום תקרות שעושים מנסרים שרוחבם פחות מטפח. (עוד אפשר לומר, שפחות משיעור טפח הוא שיעור שאין בו שום חשיבות...)
However, they did not write until what width one ought to be stringent. Nevertheless, it is agreed that it is permitted to use planks narrower than a tefach for sechach, and that is how they were accustomed to rule in Jerusalem ... In practice, nowadays there are no roofs made out of planks narrower than a tefach. (One can also suggest that [an object of] less than a tefach is of a dimension which has no significance...)