Turns out it means “to suppose, to expect.”
(Ogoru Heike wa hisashikarazu; “The haughty Heike are not for long”)
Pride comes before a fall, and outright arrogance is especially vulnerable. The more a person becomes full of themselves, and allows themselves to preen and swagger and look down on others, the more likely they are to be taken down by their own overreach, or by the enemies they’ve made. A warning to be all the more circumspect when your star is in the ascendant.
We begin with the verb 驕る (ogoru). These days it’s almost exclusively used to mean “to treat somebody,” but this is derived from an older usage, “to be (excessively) proud,” by way of expressing one’s pride through extravagant spending. The verb appears in prenominal form, attaching to and describing the proper noun 平家 (Heike), a common term for the Taira clan. (The compound can be read literally as the House = 家 of the Taira = 平.) This proper noun is marked by the particle は (wa) as being the topic of discussion, and the entire comment on this topic is a single adjective, 久し (hisashi), “long [time].” The しから (shikara) ending is an alternate version of the imperfective form, and is followed by negative suffix ず (zu) in conclusive form.
As you’d expect, this comes from a passage in the Tale of the Heike (平家物語, Heike monogatari): 「驕れる人も久しからず、ただ春の夜の夢のごとし」 (emphasis mine), Ogoreru hito mo hisashikarazu, tada haru no yo no yume no gotoshi – “The proud person is not for long, like unto a mere dream on a night in spring.”
A variant phrase replaces the specificity of the Heike with 驕る者 (ogoru mono), “a proud person.” Another, bleaker variant says that 驕る平家に二代なし (ogoru Heike ni nidai nashi), “the proud Heike have no second generation.” Given its cultural resonance, it should be no surprise that the Heike are associated with myriad sayings along these lines.
These days the character 驕 is a bit rare and one might be tempted – or pushed by an IME while typing – to use a homophone such as 奢 (a near-synonym referring to “extravagance”), but this is considered an error, presumably because it doesn’t accord with the original kanji usage.
(“Gii-kun wa… IT gyoukai ni haitta toki, ookanemochi ni natte mo jimi na seikatsu wo okuru tsumori da tte itteta no ni, ogoru Heike wa hisashikarazu no you ni choushi ni nori makutte kara sukyandaru kara nukedasenai joutai ni ochiitta sou da.”)
[“Gii-kun… when he went into the IT world, he said that even if he got super rich, he’d lead a low-key lifestyle. But despite that, he went around getting carried away all over the place, and like how the arrogant Heike were short-lived, it seems he’s mired in scandal.”]