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The phrase "almost universally accepted" is correct and usable in written English.
It can be used to describe something that is widely recognized or agreed upon by a large majority, though not necessarily by everyone.
Example: "The theory of evolution is almost universally accepted among scientists as the best explanation for the diversity of life on Earth."
Alternatives: "widely accepted" or "generally acknowledged".
Exact(59)
There are a number of factors that theorists take to provide such constraints: some will be almost universally accepted; others will have different rates of acceptance.
Special relativity, first introduced by a clerk in a Swiss patent office 100 years ago and now almost universally accepted, presents an exemplar of the "block universe".
The U.S. dollar is almost universally accepted in Afghanistan.
It is almost universally accepted that many coaches have wild mood swings.
By the end of the 19th century, physicists almost universally accepted the wave theory of light.
The strong United States dollar (worth $1.56 Canadian at current exchange rates) is almost universally accepted.
By that point, it was almost universally accepted — most of all by Stern — that their financial status quo was unsustainable.
But for most Thais, this ancient belief in a powerful world of spirits and ghosts is almost universally accepted as a part of daily life.
It is now known, or at least almost universally accepted, that glacial cycles are initiated by slight, periodic variations in the earth's orbit.
This conceptual framework in which scientists now understand the evolution of Earth's lithosphere termed plate tectonics—is almost universally accepted, although many details remain to be worked out.
Similar(1)
Of course, there are those in the industry – I suspect in about the same proportion as in all walks of life – who question the almost universally-accepted orthodoxy.
More suggestions(15)
soon universally accepted
almost entirely accepted
were universally accepted
near universally accepted
almost worldwide accepted
seems universally accepted
some universally accepted
almost universally favored
almost universally admired
almost universally viewed
almost universally criticized
almost universally liked
almost universally ratified
almost universally underfunded
almost universally praised
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