The author is unable to prove Conjecture 7.4 at present, but here are some reasons for believing it, amounting to a sketch of a partial proof. In §7.1 we started with the fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$, and extracted the functions $u_{a,b},v_{a,b}$ from it. However, any proof of Conjecture 7.4 will probably go the other way, and first construct functions $u_{a,b},v_{a,b}$ satisfying (57), and then define the fibres $N_{a,b,c}$ by (54), and put them together to form $f$.

It is not obvious that if we did define families of functions $u_{a,b},v_{a,b}$ satisfying (57), then the corresponding SL 3-folds $N_{a,b,c}$ would actually be the fibres of a fibration. We will show that part (ii) of Assumption 7.1 comes close to ensuring that they do, because it implies that the $N_{a,b,c}$ are disjoint.

A 3-dimensional family of disjoint 3-folds in $\mathbin{\mathbb{C}}^{3}$ must locally define a fibration. So if we define $U$ to be the total space of all the $N_{a,b,c}$ then we do have a fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$ with fibres $N_{a,b,c}$. Therefore, we have more-or-less reduced the problem to finding families of functions $u_{a,b},v_{a,b}$, which need only be defined near $(0,0)$ in $\mathbin{\mathbb{R}}^{2}$ for small $a,b$, satisfying Assumption 7.1 and parts (i)–(vi) of Assumption 7.1.

Now by Proposition 6.4, given any real analytic values for $u_{a,b}$ and $v_{a,b}$ on the $x$-axis, there exist unique solutions of (57) near the $x$-axis with these values, except when $a=0$ near points $(x,0)$ with $u_{0,b}(x,0)=0$. But part (v) of Assumption 7.1 gives $v_{a,b}(x,0)\equiv 0$. Thus the function $u_{a,b}(x,0)$ captures all the essential information about the behaviour of $u_{a,b}$ and $v_{a,b}$ near the $x$-axis.

Note also that part (ii) of Assumption 7.1 can be restricted to the $x$-axis. For if $u_{a,b}(x,0)<u_{a,b^{\prime}}(x,0)$ for all $a,b,b^{\prime},x$ with $b<b^{\prime}$, then by continuity of the $u_{a,b}$ we have $u_{a,b}(x,y)<u_{a,b^{\prime}}(x,y)$ for sufficiently small $y$. Thus part (ii) holds near the $x$-axis, so by making $U$ smaller if necessary we ensure that part (ii) of Assumption 7.1 holds on all of $V_{a,b}$.

We may therefore try to proceed as follows. We choose real analytic functions $u_{a,b}(x,0)$ satisfying the applicable parts of Assumptions 7.1 and 7.1, in a fairly arbitrary way. We then extend these uniquely to satisfy (57) in a small open neighbourhood of the $x$-axis. Finally we construct the associated SL 3-folds, and argue that they locally form a fibration of $\mathbin{\mathbb{C}}^{3}$.

The main problem with this approach is when $a=0$ near the singular points $(x,0)$ with $u_{0,b}(x,0)=0$, as Proposition 6.4 does not apply. To overcome this problem we will need a good understanding of how solutions of (32) behave near isolated singular points of order 1 and 2. If we could prove Conjecture 6.14, then probably we could prove Conjecture 7.4 too.