This is a very interesting design found in the Yuzuru Hiraga archives of the University of Tokyo:
https://iiif.dl.itc.u-tokyo.ac.jp/repo/s/hiraga/document/f2ea12ca-8f18-426c-ac09-0a1cdc4cf822#?c=0&m=0&s=0&cv=1&xywh=-659%2C0%2C8261%2C4927
It shows a light long thin warship with 3 single 30cm Cannons with accompanying 14cm Guns mostly in casemates. While the design shows resemblance to the Royal Navy's Glorious, Courageous and Furious class light Battlecruisers there is a significant difference.
The RN ships were capital ships with capital ship like hulls and similar superstructures while this Design X was of cruiser heritage evident of her shape but most notably it's bridge structure! Because of this, this proposal is more closely relate to the USN's Battlescout idea from around the same time which too carried similar weapons in similar numbers on long sleek fast hulls. There are no design date stated but based on the secondary guns of 14cm calibre but especially of the 8cm (76mm) AA gun calibre together with the 53cm torpedo tubes it is surely a pre 1919 proposal, likely from around 1915. Because after WW1 the new standard AA calibre of the IJN was the 12cm and the standard torpedo was of the 61cm.
The design with it's light armour (within the standard of light cruisers) and lighter armament meant a much cheaper design for either coastal bombardment or battefleet scouting operations compared to her American and British counterparts.
The design had the following characteristics:
Dimensions: 201,17m (wl) x 18,9m x 6,09m
Displacement: 11.000tons (standard)
Armour: 25mm Deck, 76mm Belt
Engines: 135.000shp Gihon or Kampon Steam Turbines, 4 Shafts
Speed: 67km/h (36knots)
Armaments:
3x1 30cm/45 Type 37 Cannons
8x1 14cm/50 Type 3 Guns
2x1 8cm/50 (76mm) Type 3 AA Guns
4x1 53cm Above water Torpedo tubes
On the revised design:
Though I have no evidence that it was developed before, during or after the Design X proposed, this 4 turreted version is somewhat of a mystery! While it indeed features more armament then the Design X, it's hull is actually shorter and even thinner being only 18m wide rather 18,9-19! While retain the same armour and tonnage! To add to this mystery on one note describing this ship mentioned not only the Glorious and Courageous of the Royal Navy but the Deutschland class Big Gun Cruisers when at that time the RN ships were long carriers! None the less I've tried to make a drawing of this design.
Another weird thing is that Yuzuru Hiraga compared or as it seems derived it's hull or calculations from the Kuma class light cruisers which were while ordered in 1917 finished only in 1920 so a design date might be closer to 1918?
Here are the archive documents describing this proposal:
https://iiif.dl.itc.u-tokyo.ac.jp/repo/s/hiraga/document/b837b31c-ea53-4c00-928d-170c38ed49ee#?c=0&m=0&s=0&cv=0&xywh=2186%2C2550%2C2561%2C1528
https://iiif.dl.itc.u-tokyo.ac.jp/repo/s/hiraga/document/42028404-f422-4e04-9fcd-668a63795968#?c=0&m=0&s=0&cv=0&xywh=-1988%2C0%2C10918%2C4927
The design had the following characteristics:
Dimensions: 199,64m (wl) x 18m x 6,06m
Displacement: 11.000tons (standard)
Armour: 25mm Deck, 76mm Belt
Engines: 135.000shp Gihon or Kampon Steam Turbines, 4 Shafts
Speed: 67km/h (36knots)
Armaments:
4x1 30cm/45 Type 37 Cannons
7x1 14cm/50 Type 3 Guns
4x1 8cm/50 (76mm) Type 3 AA Guns
6x1 53cm Torpedo tubes probably 4 side above and 2 underwater bow-stern ones