South Korea, submerged pantoon/barge test launched a Hyunmoo 4-4SLBMvariant of its Hyunmoo 2Bmissile in early July 2021 - see my July 6, 2021 report.
In early September 2021 South Korean progressed to test launching a Hyanmoo 4-4 from the first of class KSS-3 (aka KSS-lll) Dosan Ahn Changho conventional SSB submarine.
A useful report is at Reuters dated September 7, 2021.
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Pete Comment
SLBMs, all with nuclear warheads, have been developed by seven other countries, including the US, Russia, China, UK, France, India, and North Korea. It is the nuclear warhead standard for SLBMs that is one reason South Korea would likely only use nuclear warheads on its SLBMs.
Basically the KSS-3 Dosan Ahn Changho-class submarines South Korea is gradually deploying would be an extreme expense if used to merely launch high explosive SLBMs from their 6 silos. With a KSS-3 each having 6 SLBMs, but only with high explosive warheads, this would represent a high explosive throw-weight totalling only 6 tonnes. This would be less than the 8 tonnes of high explosive delivered by each of South Korea's F-35As (18 in service, so far).
SLBMs are only economical and act as a deterrent if they are nuclear armed. Otherwise North Korea could feel comfortable first or second striking South Korea with its nuclear tipped SLBMs, land based missiles and free fall (aircraft dropped) nuclear bombs against Seoul and other critical targets in South Korea. This is assuming the US may one day withdraw the nuclear deterrent "umbrella" that protects South Korea.
In any case the unwritten "Law" of Incoming Ballistic Missile Ambiguity means that North Korea must assume that all South Korean SLBMs, on their way to North Korea, would be nuclear tipped and deserve a nuclear response. Otherwise if North Korea didn't use its nuclear missiles prior to the South Korean missiles arriving North Korea could permanently lose its nuclear missiles.