This is an interesting area of a national importance (no chance to become WHS). It is popular for tourists from Slovakia and also from Czechia. It consists of two components (i) Slovenský Ráj National Park, and (ii) Zádiel canyon of Slovenský Kras NP (parts of this NP already inscribed as WHS). However, there are quite a lot of similar places even in Slovakia in Malá and Velká Fatra, Choč, Muráňská Planina mountains, to name at least some of them.

I have visited only Zádiel Canyon in the Eastern part of Slovakian Karst, not far from the city of Košice. It is a monumental structure visible from big distances. The entrance to the gorge is regulated by fee around 2 EUR. While it is pleasant walking there, I am afraid it cannot compete with canyons of other European regions (Crete, Northern Greece, Southern France etc.). Furthermore, the nearby area has been already declared as WHS.

The component of Slovenský Ráj (Slovakian Paradise) is very popular touristy area of deep & narrow limestone canyons covered by dense forests, and the entrance is also strictly regulated. However, the crowds of tourist repelled me from the visit. There are ladders and secured paths with only one-way direction allowed. Slovenská Ráj NP is located in Spiš region, thus, quite close to Spišksý hrad and Levoča WHS.

Instead of Slovenský Ráj, I enjoyed similar canyons in Fatra region – Janošíkové Diery, that are somehow similar, closer to Czechia and not so crowded.

Further updates and changes in the nomination:

I just find an article in Slovak journal Environmagazín (5/2007, p.16-17) in my archive, on which I completely forgot, and it describes the changes intended by the state party of Slovakia. It has the title “On the Nomination of Mesozoic Canyons o Western Carpathians” and explains that the former nomination from 2002 focusing mostly on the Slovenský Ráj had been refined in 2007 to increase the chanced for the inscription. More components were added - all together 12 canyons in Slovakia:

(1-2) Blatnická and Gaderská dolina in Velká Fatra mountains

(3-4) Prosiecká and Kvačianská dolina in Chočské Vrchy mountains

(5) Jánska dolina in Nízké Tatry mountains

(6-9) Piecky, Velký Sokol, Sokolia dolina, Kysel in Slovenský Ráj NP

(10-11) Zadielska and Hájska dolina in Slovenský kras NP

(12) Prielom Dunajca in Pieniny mountains. 

From this enlarged list I have visited not only already described Zadiel and Pieniny (which had been likely intended as the separate nomination with Poland and also my review there...), but also Blatnicka and Gaderska dolina in very beautiful Velka Fatra mountains (PHOTO). These canyons of Velká Fatra mountains are quite easily accesible from village Blatnice (meadows around it are the part of another TWHS of Slovakia). As evident from the Photo, walking through the canyon is not very spectacular because of the dense vegetation. I Admired these valleys much better from above - from the main ridge of the Velká Fatra mountains. The components No3-4 in Chočské vrchy are located not far from the Velká Fatra, and they are accesible from town Ružomberok. I could see the mountains from distance but have not visited yet.
However, my opinion on this TWHS has not changed…