I hope so. Because if people start using more the LN than the main chain for any kind of transaction, then miners would be doomed, IMO. But, it seems very unlikely for that to happen since the LN would make transactions on the main chain for opening/closing channels. Therefore, in the end, miners will always get their block reward and fees as usual.
And miners could also open well-connected LN nodes, integrate them into channel factories charging a small fee for "recharging" channels etc. . It is short-sighted if miners oppose LN because they fear a smaller income stream from transaction fees, because if LN is a success, then the growing user base will compensate them for transactions which go off-chain (which already exist, albeit via centralized services like Coinbase). Even Jihan Wu once wrote on Twitter, "miners love LN".
The reason why I think that LN will not replace on-chain tx completely is more related to the security concept of LN, which in my opinion isn't really suited for transactions where your life depends on (e.g. your salary). Instead, I expect people using LN as a "prepaid card", recharging their channels (via LN/channel factories) with e.g. 100-500$ per month and then doing all the small transactions via this method.
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Well, if LN is as secure as Bitcoin's main chain, then there should be nothing to worry about. After all, opening/closing a channel in the LN, requires an on-chain transaction to be made. As such, the LN would obtain the same level of security as the main chain since it requires the blockchain to settle HTLCs. However, as with any new technology, it needs to be properly tested, for issues to be addressed, making it as resilient as possible against external attacks. 
The difference is that because LN transactions can be "reversed", there are two things to consider:- you must always have a backup of all the transactions you interchanged with the other side of the channel (and with all sides of a channel factory)
- you must be online at least so often that nobody can reverse your channel state. This is, however, what "watchtowers" want to address.
It is however possible, that if a large hub wants to defraud most of its users, it could combine their LN attack with some other (double-spending?) attack, which could be dangerous if the hub is also a big mining pool. So for me it's crucial that the community should be wary that hubs do not grow to a size of more than a certain number (e.g. 10K) of users, and if possible, to use routes which avoid them. (It may not be entirely possible to figure out which hubs are operated by the same operator, which could be a problem. Need to think a bit about that. )