Zuck got the fighting arena installed in his backyard this week ... and when he pointed it out to his significant other, Priscilla Chan, he appeared to think she'd be stoked about him being able to now grapple just a few feet from his living room.

"It looks awesome," he said over text to Chan ... adding, "We have plenty of yard space!" when he noticed she wasn't exactly thrilled over it.

That's when, however, she dropped a bomb on him ... saying, "I have been working on that grass for two years."

Seems Zuck knows he now needs to at least consider getting rid of his new training toy ... because he posted a poll asking his followers whether he should "Keep the octagon" or "Save the grass."

For now, the masses believe he should say bye-bye to the sod ... 'cause, after all, dude might be fighting Musk in just a matter of months.