Depending on Bitcoin's price in the future, miners could either continue to support the Blockchain or abandon it altogether. After all, there will be no more newly minted coins which means that miners will live off TX fees.
The primary factor that determines the price of bitcoin is the law of demand and supply. Assuming that by the year 2140 all the bitcoins are already mined, supply will decrease while demand will increase resulting to a significant increase on its price. Now that there are no bitcoins no mine, miners would simply maximize their profits from transaction fees.
Currently, the price of each transaction depends upon the current price of bitcoin in the market. I doubt that miners would abandon their work but expect a fewer people engaging into this kind of business.
My thinking is transaction fees will incentivize miners to secure the network once all 21 million have been mined, which to me implies a very high value to justify the energy expenditures.
That is actually true. No more coins to mine but the compensation would shift to the transaction fees paid. Assuming that the price of bitcoins increase in the future (which would inevitably happen), transaction fees would be their saving glory in order to compensate high energy fees.