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竟 and 境 are both read ging2 in Cantonese, which usually corresponds to a third tone syllable in Mandarin (as is the case for 警, which is ging2 in Cantonese and jing3 in Mandarin).
However, both of them end up as jing4 in Mandarin. This table which gives sound correspondences from the Middle Chinese tones indicates it should be in tone class 3, but the Mandarin reflex is not third tone as expected.
點解呢? (为什么呢?)
As your linked table indicates, the Middle Chinese 陰上 tone generally corresponds to Cantonese tone 2 and Mandarin tone 3, so it is indeed curious that you see both words having tone 4 in Mandarin, which typically corresponds to Middle Chinese 去 tones or 陽上 tones where the syllable onset is an obstruent (全濁聲母).
Looking up the characters in the Kangxi Dictionary 《康熙字典》 appears to hint at a possible explanation:
竟
【廣韻】【韻會】【正韻】𠀤居慶切，音敬。窮也，終也。
From the Guangyun, Yunhui, and Zhengyun: Initial of 居; final and tone of 慶; pronounced 敬. Means limit, final.
【集韻】舉影切，音景。與境同，界也。
From the Jiyun: Initial of 舉; final and tone of 影; pronounced 景. Means the same as 境, boundary.
境
【唐韻】【正韻】居影切。
From the Tangyun and Zhengyun: Initial of 居; final and tone of 影.
【集韻】【韻會】舉影切，𠀤音景。
From the Jiyun and Yunhui: Initial of 舉; final and tone of 影; and pronounced 景.
[BTW, if you're wondering what 𠀤 is, it's a variant of 並.]
The entry for 境 indicates that it has a 上 tone (since 影 and 景 both have 上 tones), as expected; however, it also indicates that the pronunciation of 竟 in its primary meaning has a 去 tone (since 慶 and 敬 both have 去 tones). Only when 竟 is used as a substitute for 境 is it pronounced with a 上 tone.
From this, we should expect that 竟 is pronounced ging3 in Cantonese and jing4 in Mandarin and that 境 is pronounced ging2 in Cantonese and jing3 in Mandarin; however, because 竟 can be used as a substitute for 境, perhaps their tones ended up getting mixed up, with Cantonese settling on ging2 and Mandarin settling on jing4 for both characters? This is purely speculation on my part, but I haven't found any other sources that explain this.
