We know nothing specific about the equipment of the Spartans at Thermopylai. All our sources say is that they wore more armour and carried longer spears than their Persian counterparts, which doesn't actually give us any specifics. The best we can do is look at evidence from near-contemporary vase paintings, sculpture and the like, and assume that Spartan practices more or less followed suit with those in the rest of mainland Greece.

That said:

there was a transition from the classic Corinthian helmet to the more pikeman like polis at the 5th century mark.

I'm not sure what you mean by "pikeman-like polis". The Macedonian phalangite appeared more than a century after Thermopylai, but I assume that's not what you're getting at. I assume you're referring to the broad process we call the lightening of the panoply, which may or may not characterise the development of hoplite equipment from the late 6th to the 4th century BC. I've written about this process in more detail here. We don't really know the reason behind this, and we don't know if it was a real change or just a change in the way hoplites were depicted. But we can say for sure that there is a sharp drop in the amount of extant bronze armour after the 6th century BC, which probably reflects that it was used less widely than before.

Around the time of the Persian Wars, we're still fairly early in this process. Bronze armour (of the muscle cuirass type than the older bell cuirass) would still have been worn by elite warriors, along with the lighter, more widespread linothorax, a cuirass of quilted or glued linen. Some hoplites may still have worn the heavy Corinthian helmet, although lighter forms like the Attic and Thracian type were coming into vogue. Greaves would still have been fairly common, though they would go almost entirely out of style in mainland Greece in the course of the next century.

Specific to the Spartans, we really only have two fixed points: the statue you've linked to, which dates to the middle of the fifth century and shows a hoplite with a muscle cuirass and Attic helmet; and Thucydides' account of the battle of Sphakteria (425 BC), where he notes that the Spartans wore pilos helmets (conical and open-faced). The latter is assumed to be the standard form of hoplite headgear by c. 400 BC, with few exceptions. These two points seem to fit the general development of hoplite armour in the wider Greek context, so we can assume that Spartans of the Persian Wars wouldn't have looked much different from the Athenians and others that we see on vases - especially since there is no evidence of Spartans adopting their famous red tunics and lambda shields as early as the Persian Wars.

So, to answer your question, we should probably assume that the Spartan hoplites at Thermopylai would have worn a mixture of Corinthian, Attic and possibly Thracian helmets. However, since the Spartans seem to have been somewhat militarily backward at this time compared to other Greeks, it is possible that many of them stuck to the old Corinthian helmet where Greeks of other states were moving on to lighter varieties. I have no idea where the idea comes from that the Spartans would have preferred open-faced types, but if there is any source that says this, it would undoubtedly be referring to the 4th century BC, when Spartan military practice had overtaken its rivals.