You have a misunderstanding about the capacitor.
It sort of limits the current through the LEDs. (I would Not trust it)
It does Not solve the problem of reverse voltage on the LEDs.
Standard White LEDs have a forward voltage of about 3.1V The reverse voltage rating is 5V, meaning more than 5V backwards blows them up.
If the forward voltage is 3.1V on the LEDs pointing to the right, then the reverse voltage applied to the LEDs pointing left is 3.1V.
The reverse voltage is kept below 5V because the "other" LEDs keep it at 3.1V.
Do Not trust the data sheets regarding the white forward voltage at 3.4V. It is a sure way to ruin the LEDs.
V RMS is something to consider. 126VAC RMS line voltage peaks at 178V. It is almost like 178VDC for a micro-second. I believe that is why most home-brew high voltage AC long strings of LEDs fail.
Consider going all the way back to the basics if you want the LEDs to last for years instead of weeks or months.
I would recommend finding a 12 to 18V transformer (16V door bell transformer?), rectify and filter the output to DC, design the circuit with more accurate numbers, test the real ma in the circuit, and keep each LED under 18ma.
Or a switching power supply (laptop?), then repeat above.
I have a feeling most of the online LED circuits were tested for a day or "a whole week!" before the circuits were published.
If I built the circuit shown in the pdf, I would reduce the cap value to 0.22uF, change the 1K resistor to 3.9K, and remove the jumpers between reversed LED pairs.
I would want the AC current reading to be under 14ma, probably about 12ma.
Then test it. A Lot. Before making any changes.
G-