liblwjgl64.so is indeed there, and it is readable. It does not appear to be a permissions issue.
As far as I understand, the actual issue has to do with libjawt.so which does not seem to exist anywhere.
More complete terminal output for you:
[user@localhost ~]$ java -jar Downloads/Tales\ of\ Androgyny\ Linux64\ v0.1.15.1/TalesOfAndrogyny.jar
LwjglApplication: Couldn't initialize audio, disabling audio
java.lang.UnsatisfiedLinkError: /tmp/libgdxuser/31ce78a2/liblwjgl64.so: libjawt.so: cannot open shared object file: No such file or directory
at java.lang.ClassLoader$NativeLibrary.load(Native Method)
at java.lang.ClassLoader.loadLibrary0(ClassLoader.java:1941)
at java.lang.ClassLoader.loadLibrary(ClassLoader.java:1824)
at java.lang.Runtime.load0(Runtime.java:809)
at java.lang.System.load(System.java:1086)
at org.lwjgl.Sys$1.run(Sys.java:70)
at java.security.AccessController.doPrivileged(Native Method)
at org.lwjgl.Sys.doLoadLibrary(Sys.java:66)
at org.lwjgl.Sys.loadLibrary(Sys.java:87)
at org.lwjgl.Sys.<clinit>(Sys.java:117)
at org.lwjgl.openal.AL.<clinit>(AL.java:59)
at com.badlogic.gdx.backends.lwjgl.audio.OpenALAudio.<init>(OpenALAudio.java:72)
at com.badlogic.gdx.backends.lwjgl.LwjglApplication.<init>(LwjglApplication.java:86)
at com.badlogic.gdx.backends.lwjgl.LwjglApplication.<init>(LwjglApplication.java:67)
at com.majalis.traprpg.desktop.DesktopLauncher.main(DesktopLauncher.java:43)
java.lang.NoClassDefFoundError: Could not initialize class org.lwjgl.Sys
at org.lwjgl.opengl.Display.<clinit>(Display.java:135)
at com.badlogic.gdx.backends.lwjgl.LwjglGraphics.setVSync(LwjglGraphics.java:557)
at com.badlogic.gdx.backends.lwjgl.LwjglApplication$1.run(LwjglApplication.java:122)
Exception: java.lang.NoClassDefFoundError thrown from the UncaughtExceptionHandler in thread "LWJGL Application"
[user@localhost ~]$ cat error.txt
java.lang.NoClassDefFoundError: Could not initialize class org.lwjgl.Sys
at org.lwjgl.opengl.Display.<clinit>(Display.java:135)
at com.badlogic.gdx.backends.lwjgl.LwjglGraphics.setVSync(LwjglGraphics.java:557)
at com.badlogic.gdx.backends.lwjgl.LwjglApplication$1.run(LwjglApplication.java:122)
[user@localhost ~]$ ls -lah /tmp/libgdxuser/*
/tmp/libgdxuser/31ce78a2:
total 856K
drwxrwxr-x. 2 user user 80 Jan 31 18:22 .
drwxrwxr-x. 4 user user 80 Jan 31 18:22 ..
-rwxrwxr-x. 1 user user 459K Jan 31 18:22 liblwjgl64.so
-rw-rw-r--. 1 user user 395K Jan 31 18:22 libopenal64.so
/tmp/libgdxuser/6ef2d459:
total 256K
drwxrwxr-x. 2 user user 60 Jan 31 18:22 .
drwxrwxr-x. 4 user user 80 Jan 31 18:22 ..
-rw-rw-r--. 1 user user 253K Jan 31 18:22 libgdx64.so